Stellar spectropolarimetry with retarder waveplate and beam splitter devices

Transcription

1 Stellar spectropolarimetry with retarder waveplate and beam splitter devices S. Bagnulo Armagh Observatory, College Hill, Armagh BT61 9DG, U.K. and M. Landolfi INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I Firenze, Italia. landolfi@arcetri.astro.it and J. D. Landstreet Armagh Observatory, College Hill, Armagh BT61 9DG, Northern Ireland, U.K. Physics & Astronomy Department, The University of Western Ontario, London, Ontario, Canada N6A 3K7 jlandstr@astro.uwo.ca and E. Landi Degl Innocenti Dipertimento di Astronomia e Scienza dello Spazio, Università degli Studi di Firenze, Largo E. Fermi 5, I Firenze, Italia. landie@arcetri.astro.it and L. Fossati Institut fuer Astronomie, Universitaet Wien, Tuerkenschanzstrasse 17, A-1180 Wien, Austria. fossati@astro.univie.ac.at and M. Sterzik European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile.

2 2 ABSTRACT Night-time polarimetric measurements are often obtained very close to the limits of the instrumental capabilities. It is important to be aware of the possible sources of spurious polarization, and to adopt data reduction techniques that best compensate for the instrumental effects intrinsic to the design of the most common polarimeters adopted for night-time observations. We define a self-consistent framework starting from the basic definitions of the Stokes parameters, and we present an analytical description of the data reduction techniques commonly used with a polarimeter (consisting of a retarder waveplate and a Wollaston prism) to explore their advantages and limitations. We first consider an ideal polarimeter in which all optical components are perfectly defined by their nominal characteristics. We then introduce deviations from the nominal behaviour of the polarimetric optics, and develop an analytical model to describe the polarization of the outgoing radiation. We study and compare the results of two different data reduction methods, one based on the differences of the signals, and one based on their ratios, to evaluate the residual amount of spurious polarization. We show that data reduction techniques may fully compensate for small deviations of the polarimetric optics from their nominal values, although some important (first-order) corrections have to be adopted for linear polarization data. We include a detailed discussion of qualitychecking by means of null parameters. We present an application to data obtained with the FORS1 instrument of the ESO VLT, in which we have detected a significant amount of cross-talk between circular and linear polarization. We show that this cross-talk effect is not due to the polarimetric optics themselves, but is most likely caused by spurious birefringence due to the instrument s collimator lens. Subject headings: Data Analysis and Techniques Astronomical Techniques 1. Introduction Most astronomical observations are based on measurements of intensity, either per unit of time, or wavelength, or both. Polarimetry measures something more, i.e., how the electric field vector oscillates. Polarization arises every time there is a mechanism that breaks the symmetry within the radiative source, or between the source and the observer, for example the presence of a magnetic field, the presence of collimated beams, or scattering by particles. For this reason, polarimetry is a very powerful remote sensing tool for the study of a variety of objects: from asteroids and other objects of the solar-system, where the polarized nature of the reflected light gives information about

3 3 the surface structure (e.g., Penttilä et al. 2005), to magnetic stars, the spectral lines of which are polarized by the Zeeman effect (e.g., Mathys 1989), to AGB and post-agb stars (e.g., Bieging et al. 2006) and supernovae (e.g., Wang & Wheeler 2008), where polarized light may reveal asymmetries of the ejected materials. This list is clearly not exhaustive. Note that, in the context of this work, we deliberately neglect applications to solar physics. Solar astronomers often have access to observing facilities where both telescope and instrumentation were specifically designed to be aimed at high-precision polarimetric measurements, and data reduction techniques have been developed specific to each optical design. Example of dedicated instruments for solar polarimetry include, e.g., ZIMPOL (Zürich Imaging Polarimeter; Powell 1995), and SPINOR (Spectro-Polarimeter for INfrared and Optical Regions; Socas Navarro et al. 2006). In night-time astronomy, polarimetric devices are generally much simpler than the high-precision instruments that have been built for solar observations. Polarimetric modules employed in night-time astronomy may even be introduced as afterthoughts, after the instrument has been built. Even in the most favorable cases where an imager or a spectrograph has been originally designed to accommodate polarimetric optics, night-time astronomers still have to face the fact that telescopes are rarely built taking into account the most basic requirements for high-precision polarimetric observations. Some of the middle-size telescopes, such as the ESO New Technology Telescope (NTT) and the Telescopio Nazionale Galileo (TNG) at La Palma, offer instruments with polarimetric capabilities installed at the Nasmith station, since they do not have a Cassegrain focus. Nevertheless, with some of the instruments available at the Cassegrain focus (e.g., FORS at the ESO VLT), night-time polarimetric observations can be obtained with an accuracy of a few units in 10 4, which is required, for instance, to search for very small polarization features in solar-system bodies (Boehnhardt et al. 2004) or weak magnetic fields in various kinds of stars (e.g., Aznar Cuadrado et al. 2004; Aurière et al. 2007; Wade et al. 2007). Most of these polarimetric devices are based on a modulator that can be set at a series of fixed positions (e.g., a retarder waveplate or a Fresnel rhomb) followed by a Wollaston prism (or, in case of the ISIS instrument, a simple piece of calcite). For a general introduction to polarimetric techniques used in night-time astronomy, we refer to Hough (2005) and references therein. The aim of this paper is to characterize the observing strategy, data reduction, and quality-check techniques that best suit the most common kinds of polarimeter used in night-time astronomy. The discussion is aimed at the typical user who does not know the Müller matrices that fully describe the system telescope+instrument, but who needs to know what is the optimal number of positions of the retarder waveplate to adopt for the observations, how to optimally combine the data and test it for systematic errors, and how to use polarimetric standard stars for accurate data calibration. Polarimetric quantities are usually defined adopting the symbols Q, U (for linear polarization), and V (for circular polarization), but even a superficial look at literature data shows that such symbols are not uniquely associated to identical measurable quantities (e.g. Clarke 1974). Sign ambiguities are a common issue, but in some cases linear polarization measurements are given without even specifying the reference system. Ambiguity affects not only scientific papers but also instrument manuals, in which case sign indeterminacy is propagated through the entire data reduction process. The practical workaround is a comparison of newly obtained measurements with

4 4 published polarimetric data, a process that may have already been iterated more than once, bringing back sign or direction determination to a few ancient publications. Therefore, in an attempt to clarify the situation, and in spite of some redundancy with previously published material, we start with the definitions of the polarimetric quantities and with the basic concepts that underlie the design of polarimetric devices (Sect. 2). This paper will focus on optical polarimetry performed with a retarder waveplate followed by a beam splitter device, such as a Wollaston prism. We start with the ideal case (Sect. 3), and then go on to analyze the impact of deviations from the ideal case (Sects. 4 and 5). We describe the use of the null parameters as a quality-check tool for data reduction (Sect. 6). Our analytical model does not include a treatment of spurious retardation or polarization introduced by the optics that precede the polarimetric module (e.g., telescope optics). We will not discuss problems introduced by fringing, which often affects super-achromatic waveplates, and for which we refer to Aitken & Hough (2001). Note that fringing problems may be minimized by using Fresnel rhombs instead of waveplates, and that most of the results presented in this paper can be easily extended to instruments that employ this kind of retarder. Our presentation of theoretical considerations concerning data analysis is then applied to observational data obtained on the sky with the FORS1 instrument of the ESO VLT (Sect. 7). 2. Definitions The polarization properties of a light beam are described by the specification of four independent quantities that were introduced by Stokes (1852) and which are commonly referred to with the symbols I, Q, U, and V. Their definitions require the preliminary choice of a reference direction pertaining to the plane perpendicular to the direction of propagation of the radiation. We will refer to a right-handed coordinate system, (x, y, z), with the z-axis directed along the direction of propagation and the x-axis directed along the reference direction. In stellar astronomy, the reference direction is generally taken as the celestial meridian passing through the observed object. In polarimetric measurements of solar-system objects (e.g., asteroids) it is common to adopt as the reference direction the great circle passing through the object itself and the Sun, or, as a better choice, the direction perpendicular to it (see, e.g. Bagnulo et al. 2006b). According to the Maxwell equations, the electric field vector of the radiation beam lies in the (x, y) plane so that it is described by only the two components E x and E y. In a fixed point of space along the direction of propagation, these two components of the electric field are described by the functions of time, E x (t) and E y (t): ) E x (t) = R (Ẽx (t)e iωt ) (1) E y (t) = R (Ẽy (t)e iωt where Ẽx(t) and Ẽy(t) are slowly varying complex functions of the point (x, y) and of time t, i is

5 5 the imaginary unit, and the symbol R( ) means the real part of ( ). We introduce the Fourier transforms, E x (ω) and E y (ω), by the definition E x (ω) = E y (ω) = + + E x (t)e iωt dt E y (t)e iωt dt. (2) For the quasi-monochromatic case, the Stokes parameters of the radiation beam of angular frequency ω are defined by the expressions (Landi Degl Innocenti et al. 2007): I(ω) = k [E x (ω) E x (ω)+e y (ω) E y (ω)] Q(ω) = k [E x (ω) E x (ω) E y (ω) E y (ω)] U(ω) = k [E x (ω) E y (ω)+e y (ω) E x (ω)] V (ω) = ik [E x (ω) E y (ω) E y (ω) E x (ω)] (3) where the symbol means complex conjugate, and k is a dimensional positive constant, irrelevant whenever absolute measurements are not important. As a matter of fact, we always use relative quantities, that is, we normalize Q, U, and V to the intensity, and we consider the reduced Stokes parameters P Q, P U, and P V defined by P Q = Q/I P U = U/I P V = V/I. (4) According to the definition given by Eqs. (3), Stokes Q is the difference between the intensity of the radiation with the electric field oscillating along the reference direction and the intensity of radiation with the electric field oscillating in the direction perpendicular to it; Stokes U is the difference between the radiation intensity with an electric field oscillating at 45 and the intensity with electric field oscillating at 135 with respect to the reference direction (angles are reckoned counterclockwise from the reference direction looking at the source). Stokes V is given by the so-called right handed circular polarization minus the left handed circular polarization.these are defined such that at a fixed point of space, the tip of the electric field vector carried by a beam having right-handed (or positive ) circular polarization rotates clockwise, as seen by an observer looking at the source of radiation. Conversely, the tip of the electric field vector of a beam having left-handed (or negative ) circular polarization rotates counterclockwise, as seen by an observer looking at the source. These definitions are consistent with those given by Shurcliff (1962).

6 Ambiguities intrinsic to the definition In the literature, there are several different possible definitions of Stokes parameters, that come for instance from different assumptions about the reference system (e.g., in radio astronomy the reference system is often chosen left handed), or from the definition of the electric field vector, as in Eq. (1), where one could use the + sign instead of in the exponential. For more details, we refer to the papers by Clarke (1974) and by Landi Degl Innocenti et al. (2007). Using the Fourier transform, we have introduced a further subtle ambiguity arising from the definition of Fourier transform, as Eq. (3) implicitly adopts for the Fourier transform the representation of Eq. (2). Using the alternative definition with the substitution e iωt e iωt namely, defining E x (ω) = E x(t)e iωt dt E y(ω) = E y(t)e iωt dt one obviously has, since E x (t) and E y (t) are real, (5) E x (ω) = E x(ω) E y (ω) = E y(ω) (6) so that the sign of the V Stokes parameter changes when the definition of the Fourier transform is changed from Eq. (2) to Eq. (5). In practice, this means that when adopting Eq. (3) for the definition of the Stokes parameters, one has to be aware that the definition implies a convention about the choice of positive or negative time in the definition of the Fourier transform. We note that a similar ambiguity is present also when adopting traditional definition based on spatial coordinates, due to the arbitrary choice in the imaginary exponent used to represent the electric field vector. For more details, see Eqs. (1.11) and (1.12) of Landi Degl Innocenti & Landolfi (2004) Fraction of linear polarization and position angle Linear polarization is often expressed also in terms of P L and Θ, where P L is the fraction of linearly polarized radiation, and Θ is the angle of maximum polarization, i.e., the angle that the major axis of the polarization ellipse forms with the x-axis of the reference system, reckoned counterclockwise for an observer looking at the radiation source: P L = PQ 2 + P U 2 P Q = P L cos(2θ) P U = P L sin(2θ). (7)

7 7 The correct expression that gives the position angle is Θ= 1 ( ) 2 arctan PU +Θ 0 (8) P Q where or 0 if P Q > 0 and P U 0 Θ 0 = 180 if P Q > 0 and P U < 0 90 if P Q < 0 Θ= { 45 if P Q = 0 and P U > if P Q = 0 and P U < 0. (9) 2.3. Transforming Stokes parameters into a new reference system If a new reference direction is obtained from the old one by a counterclockwise rotation (looking at the source) by an angle χ, by means of Eq. (3) one can easily show that the new reduced Stokes parameters, (P Q,P U,P V ), are connected to the old ones, (P Q,P U,P V ), by the following transformation: P Q = cos (2χ) P Q + sin (2χ) P U P U = sin (2χ) P Q + cos (2χ) P U (10) P V = P V. 3. Measuring Stokes parameters with an ideal polarimeter There are several ways to realize a polarimeter (see, e.g., Serkowski 1974). In this paper we will consider the case of a beam-splitting polarimeter, i.e., built with a retarder waveplate followed by a Wollaston prism. This design has been proposed, e.g., by Appenzeller (1967), and implemented in several instruments, e.g. FORS1 of the ESO VLT (Appenzeller et al. 1998). In this Section we give a theoretical treatment of the ideal case Ideal filters for linear and circular polarization A filter for linear polarization (called a linear polarizer) is a device that can be inserted into a beam of radiation and which, by definition, is totally transparent to the component of the electric field along a given direction, perpendicular to the direction of propagation (the transmission axis of the polarizer), and totally opaque to the component of the electric field in the orthogonal direction. Waveplates are optical elements with two orthogonal principal axes, one called the fast axis, and the other one the slow axis, characterized by two different refractive indices, such that a linearly polarized beam with polarization parallel to one of the principal axes is propagated without change

8 8 in its polarization state. A quarter-wave plate produces a π/2 phase retardation between the components of the electric field along the fast and slow axes. A half-wave plate produces a π phase retardation between the components of the electric field along the fast and slow axes. An ideal filter transmitting positive circular polarization can be realized by the combination of an ideal quarter-wave plate followed by a polarizer whose transmission axis is rotated counterclockwise (looking at the source of radiation) by an angle of 45 with respect to the direction of the fast axis of the plate. A common instrumental setup consists of a rotatable retarder waveplate and an analyzer fixed and with its principal axis aligned to the reference direction. Obviously, in this configuration, the ideal filter transmitting positive circular polarization is obtained by setting the fast axis of the retarder waveplate at an angle of 45 (i.e., 45 clockwise) with respect to the reference direction, looking at the source of radiation, and the ideal filter transmitting negative circular polarization is obtained by setting the fast axis of the retarder waveplate at an angle of 45 with respect to the reference direction An ideal polarimeter We define α as the angle between the reference direction and the fast axis of the retarder waveplate, counted counterclockwise from the reference direction; β is the position angle of the transmission axis of the linear polarizer counted counterclockwise from the reference direction; γ is the phase retardance introduced by the retarder waveplate. For a quarter waveplate γ = π/2; for a half waveplate, γ = π. In the geometrical scenario of Fig. 1, the detector will measure a signal (Landi Degl Innocenti & Landolfi 2004) S(α, β, γ) { 1 2 I+ [ Q cos 2α + U sin 2α ] cos(2β 2α) (11) [ ] Q sin 2α U cos 2α sin(2β 2α) cos γ + } V sin(2β 2α) sin γ. We now adapt this general equation to our particular case, observing that for the measurement of linear polarization we use a half-wave plate (γ = π), and for the measurements of circular polarization we use a quarter-wave plate (γ = π/2). As analyzer, we will consider a Wollaston prism, which splits the incoming radiation into two beams which define an exit plane, or principal plane. The two beams are polarized, one in the direction parallel to that plane (which we will call parallel beam) and one in the direction perpendicular to it (which we will call perpendicular beam). The Wollaston prism will be inserted into the beam in such a way that its principal plane

9 9 is parallel to the reference axis. By this definition, the parallel beam and the perpendicular beam will be associated to the β values 0 and 90, respectively. We define r(α, γ) as the ratio of the signals corresponding to the parallel and perpendicular beams measured with the retarder waveplate with retardance phase γ at position angle α ( S(α, β =0 ),γ) r(α, γ) = S(α, β = 90. (12),γ) In the case of an ideal polarimeter, it is possible to see that the measurement of P Q can in principle be performed with γ = π by setting the angle α (defining the position of the fast axis of the retarder) to 0, or to any multiple of 45, whereas the measurement of P U can be performed with γ = π by setting the angle α to 22.5 or to any angle differing from 22.5 by a multiple of 45. The measurement of P V can be performed with γ = π/2 by setting the angle α to 45, or to any angle differing from 45 by a multiple of 90 : r(0,π) = 1 r(45,π) r(22.5,π) = 1 r(67.5,π) r( 45, π/2) = 1 r(45, π/2) = r(90,π) =... I+Q I Q = r(112.5 I+U,π) =... I U (13) = r(135 I+V, π/2) =... I V. By introducing we obtain, for instance, More generally, defining G(α, γ) = ( ) r(α, γ) 1 = S(α, 0,γ) S(α, 90,γ) r(α, γ) + 1 S(α, 0,γ)+S(α, 90,γ) G(α =0,γ = π) = P Q G(α = 22.5,γ = π) = P U G(α = 45,γ = π/2) = P V. G [j] Q = G(α =(j 1) 45,γ = π) G [j] U = G(α = (j 1) 45,γ = π) G [j] V = G(α = 45 +(j 1) 90,γ = π/2), (14) (15) (16) where j is an integer number, we obtain G [j] =( 1)(j 1) P (17) where stands for Q, U, or V. When dealing with real instrumentation, one can take advantage of this redundancy in the measurement of the Stokes parameters in order to compensate for various instrumental effects (such as for instance the different transparency of the two channels of the Wollaston prism). We illustrate two different methods that have been developed for this purpose,

10 10 referring to them as the difference method (e.g., FORS1/2 User manual) and the ratio method (Semel et al. 1993; Donati et al. 1997). Note that we will introduce some formalism that may appear more cumbersome than is actually needed in practice. In fact, the introduction of this formalism is necessary to deal with the deviations from the ideal case that will be discussed in Sect. 4. More easily readable formulas for the computation of Stokes parameters will be given in Appendix A, where we also provide the necessary formulas for the treatment of the photon-noise and background subtraction errors The difference method From the parallel and perpendicular beams, measured in a pair of exposures obtained at different positions of the retarder waveplate, we construct the functions and we obtain D [k] 1 2N = G[2k 1] N k=1 G [2k] (18) D [k] = P. (19) Note that N is the number of pairs of exposures that are taken at different positions of the retarder waveplate separated by 45 to measure linear polarization, or by 90 to measure circular polarization. For an even number of pairs of exposures N 2 we define the null parameters It is trivial to see that N (D) = 1 2N N (D) Q N ( 1) k 1 D [k]. (20) k=1 = N (D) U The introduction of the null parameters will be motivated in Sect. 6. = N (D) V =0. (21) 3.4. The ratio method As an alternative to the difference method, we may define r [j] Q = r(α =(j 1) 45,γ = π) r [j] U = r(α = (j 1) 45,γ = π) r [j] V = r(α = 45 +(j 1) 90,γ = π/2) (22) and R [k] = r[2k 1] /r [2k]. (23)

11 11 We obtain ( N ) 1 R [k] k=1 ) 1 R [k] k=1 ( N 2N 1 2N +1 = P (24) where, again, N is the number of pairs of observations (taken at different retarder waveplate positions). We also introduce L [k] = ( r [2k 1] /r [2k] ) Ck (25) where C k =( 1) k 1. For an even number N 2 of pairs of exposures, we define the null parameters N (R) = ( N ) 1 L [k] k=1 ) 1 L [k] k=1 ( N 2N 1 Similarly to the null parameters defined by Eqs. (20), we find N (R) Q = N (R) U 2N +1. (26) = N (R) V =0. (27) 4. First order deviations from the ideal case In this Section we will assume that a real polarimeter can be described with the same formalism used for the ideal case, but introducing small deviations from the nominal values for the angles α, β and γ used in the previous Section. We continue to consider a beam of light parallel to the optical axis of the instrument, and for this treatment we ignore all the optical elements (telescope mirrors, collimator lenses, etc) that may precede the polarimeter. A brief discussion of some of the effects that upstream optics may produce, especially for off-axis light, will be found in Sect. 7, where we discuss the FORS1 polarization optics. Nominal values α 0, β 0, and γ 0, are defined as follows: 0, 45,... =(j 1) 45 for P Q α 0 = 22.5, 67.5,... = (j 1) 45 for P U 45, +45,... = 45 +(j 1) 90 for P { V 0 for the parallel beam β 0 = 90 for the perpendicular beam { π for the half waveplate γ 0 = π/2 for the quarter waveplate. We will consider the following situations: (28)

12 12 i) a deviation δα from the nominal value α 0 ; ii) a deviation δβ from the nominal value β 0 ; iii) a deviation δγ from the nominal value γ 0 ; iv) different transmission functions in the parallel and perpendicular beam, h and h, respectively. The effects i) iii) can be formalized by adopting in Eq. (12) α, β, γ values that deviate from their nominal values and setting α = α 0 + δα δα 1 β = β 0 + δβ δβ 1 (29) γ = γ 0 + δγ δγ 1 where δα, δβ, and δγ are expressed in radians. We allow for the possibility that the deviations of the linear polarizer from the nominal values are different in the parallel and in the perpendicular beam, i.e., we assume that in the parallel beam β = δβ, and in the perpendicular beam β = 90 + δβ. We assume that the deviations δα and δγ are independent of the instrument configuration, i.e., they are constant regardless of the values of α 0, β 0, and γ 0. Finally, we introduce a coefficient δh which takes into account the difference between the transmission functions in the parallel and perpendicular beams h = h = 1 + δh (30) h and we assume δh 1. Note that by adopting Eq. (30) we make the implicit assumption that h assumes the same value at all positions of the retarder waveplate, and in particular that the images produced by the parallel and the perpendicular beam at different positions of the retarder waveplate do not change position on the detector. Later in this paper we will consider also deviations that are not small, for which we will use the symbols α, β, γ, and h. In Sect. 3 we have considered various combinations of the signal S(α 0,β 0,γ 0 ). In the following we deal with various combinations of the quantities S(α 0 +δα, 0 +δβ,γ 0 +δγ) and S(α 0 +δα, 90 + δβ,γ 0 + δγ). These measured quantities are analogous to the ideal quantities introduced in Sect. 3, and will be denoted using the symbol hat ( ). Accordingly, we introduce ˆr(α 0,γ 0 )= which is the measured analogue of Eq. (12). S(α 0 + δα, δβ,γ 0 + δγ) (1 + δh) S(α 0 + δα, 90 + δβ,γ 0 + δγ), (31) We immediately note that the use of ratios between fluxes measured at the same position angles of the retarder waveplate (which will be used in both the difference and the ratio methods presented

13 13 below) removes the effects due to changes of sky transparency between exposures, barring a strong degradation of the signal-to-noise ratio, which would prevent a proper cancellation of spurious terms. We now introduce Ĝ(α 0,γ 0 ) = (ˆr(α 0,γ 0 ) 1) / (ˆr(α 0,γ 0 ) + 1) = [S(α 0 + δα, δβ,γ 0 + δγ) (1 + δh) S(α 0 + δα, 90 + δβ,γ 0 + δγ)] / [S(α 0 + δα, δβ,γ 0 + δγ) + (1 + δh) S(α 0 + δα, 90 + δβ,γ 0 + δγ)], (32) which is the measured analogue of Eq. (14). We finally define, similarly to Eqs. (16), Ĝ [j] Q = Ĝ(α 0 =(j 1) 45,γ 0 = π) Ĝ [j] U = Ĝ(α 0 = (j 1) 45,γ 0 = π) Ĝ [j] V = Ĝ(α 0 = 45 +(j 1) 90,γ 0 = π/2). (33) By combining Eqs. (31), (32), and (33), after a tedious first order development, we obtain Ĝ [j] Q =( 1)(j 1) P Q + ( 1) j 1 4 P U δα + P Q P U (δβ δβ ) + ( 1) j P U (δβ + δβ ) P V cos ( j π 2 ) δγ 1 2 (1 P 2 Q ) δh Ĝ [j] U =( 1)(j 1) P U + ( 1) j 4 P Q δα P Q P U (δβ δβ ) + ( 1) j 1 P Q (δβ + δβ )+ 1 2 P V [ sin ( j π 2 ) cos ( j π 2 ) ] δγ 1 2 (1 P 2 U ) δh Ĝ [j] V =( 1)(j 1) P V 2P U δα + ( 1) j P U P V (δβ δβ )+P U (δβ + δβ )+ P Q δγ (1 P 2 V ) δh. (34) A comparison of the ideal expressions of G [j] given by Eqs. (17) with the corresponding measured functions Ĝ[j] given by Eqs. (34) shows that accurate measurement of the P values from an observation obtained at only one retarder waveplate position requires a fair number of calibrations. The most crucial one consists of a precise flat-fielding correction. Equations (34) show that δh = 2 %, a realistic value for the flat-fielding accuracy, will lead to 1 % of spurious polarization. A

14 14 workaround for this systematic error may be found if one is interested only in the Stokes profiles of spectral lines, and if the continuum is unpolarized. A much more satisfactory alternative is to use one or more pairs of exposures, as shown in the remaining part of this Section. For a further discussion of the case of polarimetric measurements obtained from incomplete datasets we refer to Maund (2008). We introduce 4.1. The difference method D [k] = Ĝ[2k 1] which are the measured analogues of Eqs. (18), and we define P (D) = 1 2N which are the measured analogues of Eqs. (19). To first order we obtain P (D) N k=1 Q = P Q +4P U δα P U (δβ + δβ ) ( 1 ( 1) N ) P V δγ 1 4N P (D) U = P U 4P Q δα + P Q (δβ + δβ ) P (D) V = P V P U P V (δβ δβ ). We introduce the measured null parameters number N 2 of pairs of exposures, N (D) Q = 1 2N N (D) U = 1 2N N (D) V = 1 2N N k=1 N k=1 N k=1 Ĝ[2k] (35) D [k], (36) (37) (D) N, and to first order we obtain, for an even ( 1)(k 1) D[k] ( 1)(k 1) D[k] ( 1)(k 1) D[k] All these results will be discussed in Sect Q = 1 2 P V δγ U = 0 V = 0. (38) 4.2. The ratio method Extending the treatment of Sect. 4.1, we define ˆr [j] Q = r(α 0 =(j 1) 45,γ 0 = π) ˆr [j] U = r(α 0 = (j 1) 45,γ 0 = π) ˆr [j] V = r(α 0 = 45 +(j 1) 90,γ 0 = π/2) (39)

15 15 and R [k] = ˆr[2k 1] /ˆr [2k] L [k] = ( ˆr [2k 1] /ˆr [2k] ) Ck (40) where C k =( 1) k 1. We define P (R) = ( N k=1 ( N k=1 ) 1 R [k] 2N 1 ) (41) 1 R [k] 2N +1 and we obtain, to first order, P (R) Q = P Q +4P U δα P U (δβ + δβ ) ( 1 ( 1) N ) P V δγ 1 4N P (R) U = P U 4P Q δα + P Q (δβ + δβ ) P (R) V = P V P U P V (δβ δβ ) (42) that is, the same results obtained with the difference method in Sect We also have, for an even number of pairs of exposures, N (R) Q = N (R) U = N (R) V = NQ k=1 NQ k=1 NQ k=1 NQ k=1 NQ k=1 NQ k=1 bl [k] Q bl [k] Q bl [k] U bl [k] U bl [k] V bl [k] V! 1 2N 1! 1 2N +1! 1 2N 1! 1 2N +1! 1 2N 1! 1 2N +1 = 1 2 = 0 = 0. P V δγ 1 PQ 2 (43) 4.3. Estimating Stokes parameters We conclude this Section providing some explicit and convenient formulas that relate, to first (D) (R) order, the real Stokes parameters and the measured quantities P or P. We recall that the actual Stokes parameters P, ideally representing the polarization intrinsic to radiation coming

16 16 from the source, may still possibly be contaminated by other spurious signals that come from interstellar polarization, scattered moonlight, or instrumental polarization due to any element preceding the polarimetric optics, such as for instance telescope optics or the instrument collimator. Equations (37) and (42) display the remarkable property that the difference and ratio methods give identical results both to zero and to first order. Therefore we will set P (D) (R) = P = P. (44) We first note that, both for linear and circular polarimetric measurements, combining a pair of exposures, imperfections in the flat-field corrections are suppressed to first order (the impact of a large imperfection in the flat-field correction will be discussed in Sect. 5). For the remaining deviations, we discuss linear and circular polarimetry separately Linear polarimetric measurements P Q measurements obtained with an odd number of a pairs of exposures are potentially affected by cross-talk from circular polarization, which instead cancels out if observations are performed with an even number of pairs of exposures. In Eqs. (37) or (42), the term 1 4N (1 ( 1)N ) P V δγ will be neglected in what follows, assuming that we have performed an even number of pairs of measurements. We then define δ =2δα 1 2 (δβ + δβ ) (45) and we obtain P Q = P Q + (2δ) P U P U = (2δ) P Q + P U. (46) A comparison of Eq. (46) with Eq. (10) shows that for δ 1, imperfections of the alignment of the fast axis of the retarder waveplate, and of the principal plane of the Wollaston prism (which may both be wavelength dependent) cause, to first order, a rotation in the P Q P U plane by the angle δ given by Eq. (45). Equation (46) is strictly valid under the assumption that the deviations from the ideal case are small. The most important effect on the measured Stokes parameters due to imperfections of the polarimetric optics is the term δα, i.e., the deviation from the nominal position angle of the retarder waveplate. This term appears multiplied by a factor four in Eq. (46). It is possible to obtain more accurate yet simple expressions under further reasonable assumptions. We first set δβ = δβ = δβ. This reflects a realistic situation in which the principal plane of the Wollaston prism is not perfectly aligned to the reference direction, but in which the incoming radiation is split into two beams that have a perfectly orthogonal polarization. We also set δh =0 (in Sect. 5.3 we will see that this assumption is not necessary if the ratio method is used instead of the difference method). We keep the hypothesis that linear polarization measurements are obtained

17 17 with an even number of pairs of exposures. Then we can drop the assumptions δα 1, δβ 1. For clarity, we will denote the deviations from α 0 and β 0, which may now be arbitrarily large, with α and β, respectively. We obtain: P Q = cos(4 α 2 β) P Q + sin(4 α 2 β) P U P Q = sin(4 α 2 β) P Q + cos(4 α 2 β) P U. (47) Equation (47) can be easily inverted by applying Eq. (10): P Q = P Q cos(2 ) P U sin(2 ) P U = P Q sin(2 ) + P U cos(2 ) (48) where we have set = 2 α β. (49) It appears that the measured linear polarization can be fully corrected with a simple rotation by an angle 2, for any deviation from the angles α 0 and β 0. While deviations α and β may be well calibrated in the laboratory, their combined effect may also be obtained by measuring the position angle of a standard star for linear polarization, which is expected to be wavelength independent. In summary: Eqs. (46) show that first order deviations δγ will be fully compensated using either the difference or the ratio methods, while even a small change in the orientation of the fast axis of the retarder waveplate has to be calibrated by measuring the angle of Eq. (49) as a function of wavelength, and then the measurements must be corrected using Eq. (48). It is easy to verify that the measured fraction of linear polarization is identical to first order to the actual fraction of linear polarization: P L = P L, (50) while the polarization position angle Θ is obtained from a simple offset Θ= Θ+. (51) For broadband polarimetry, a correction angle D(T F, F), analogous to for spectro-polarimetry, and pertaining to a given wavelength band and to a given source spectral energy distribution F, can be obtained by convolving (λ) with the telescope+instrument+filter response curve T F (λ), and with the model spectral energy distribution F(λ): In summary, to first order: D(T F, F) = + 0 dλ F(λ) (λ) T F (λ) + 0 dλ F(λ) T F (λ). (52)

18 18 i) Differences in the transmission functions of the two beams split by the Wollaston prism do not affect linear polarization measurements. ii) The total fraction of linear polarization is not affected by any deviation of the polarimetric optics from the ideal case. iii) Obtaining P Q, P U, and the position angle Θ from the measured quantities P Q and P U requires a process of calibration to account for first order corrections in δα and δβ. iv) δγ does not introduce any spurious effect. v) Cross-talk from P V to P Q due to the polarimetric optics can be canceled out to first order by obtaining an even number of pairs of exposures for P Q, e.g., setting α =0, 45, 90, Circular polarimetric measurements To first order, the measurement of circular polarization is a pretty robust operation. The only spurious first order contribution comes from the term (δβ δβ ) multiplied by the product P U P V, which is likely to be very small even for strongly polarized sources. This term is canceled out if δβ = δβ = δβ, in which case (to first order) we can assume P V = P V. (53) In conclusion, circular polarization can be correctly measured up to first order with a simple data reduction treatment which need not be complemented by any process of calibration. To first order, the polarimetric optics do not introduce cross-talk from linear to circular polarization. 5. Accuracy of the first order approximation In the previous Sect. we have already seen that second and higher-order effects in linear polarization measurements due to α and β deviations from the angles α 0 and β 0 are zero. We have left undefined the higher-order contributions due to a large deviation γ from the nominal retardation γ 0 of the waveplate, and those due to flat-field inaccuracies. Regarding circular polarization measurements, we have not discussed the impact of any large α, β, γ, or h deviations. The natural next step of this study is to quantify the accuracy of the first order approximation, i.e., to compare the P values obtained using a first order approximation to the ideal ones, and to identify the cases in which errors introduced by the photon noise will be negligible compared to those introduced by higher-order deviations of the polarimetric optics. (Photon noise is explicitly estimated in Appendix A.) We note that in some cases, both photon noise and higher-order effects may be negligible compared to the spurious polarization introduced by the optics that precede

19 19 the polarimetric module, and which may depend upon the telescope position while observing (a practical example will be presented in Sect. 7). The effects on linear polarization measurements due to α and β were found to be identical with both the difference and the ratio methods. For all the remaining cases, the difference and ratio methods give different results, thus in Sects. 5.1, 5.2, and 5.3 we need to re-introduce a different notation for the Stokes parameters obtained with the two methods. We will study the differences P (D) P (R) P = P and P as a function of (large) deviations α and β (for circular polarization), and of γ, and h (for linear and circular polarization) Linear polarization: effects due to γ In the framework of the difference method, it is possible to show that, to an arbitrary order, ( ) P Q = sec 2 γ (D) 2 P Q P U = sec 2 ( γ 2 ) P (D) U. (54) Equation (54) may be used for a more refined data calibration, if the quantity γ is known from laboratory measurements. However, within the context of the practical applications that are the subject of this work, Eq. (54) serves to demonstrate that, using the difference method, the expressions for linear polarization given by Eq. (48) are in fact valid up to a third-order expansion in γ. If the difference method is used, γ will lead to a systematic underestimate of the fraction of linear polarization. Equation (54) shows also that the difference method gives results that are free from cross-talk from circular to linear polarization due to the effects considered to any order. Equation (54) also expresses the (not obvious) property that the position angle of linear polarization is not affected by deviations of the retardation phase. Adopting the ratio method, an expression identical to Eq. (54) is valid only if P V = 0. In other words, linear polarization data reduced with the ratio method are also potentially affected by cross-talk from circular polarization to an order higher than one. Thus, if the ratio method is used, linear polarization measurement may under or over-estimate the actual fraction of linear polarization in a way that depends on the fractional circular polarization of the incoming radiation beam. Figure 2 shows the discrepancy P Q between the P Q values measured to first order and the actual P Q value as a function of γ as described by Eq. (54).

20 Circular polarization: effects due to α, β, and γ Using the difference method, it is possible to show that the analytical expression P V = [ sec(2 α β) sec( γ)] P (D) V (55) is valid to any order, assuming δh 1. Equation (55) may be used for a refined data calibration, if laboratory data are available, and tells us that Eq. (53) is strictly valid up to the first order only. If the difference method is adopted, the measurements will always underestimate the fraction of circular polarization. An expression identical to Eq. (55) is valid in the ratio method only if P Q = P U = 0, but without the assumption δh 1. Eq. (53) will lead to a under- or over-estimate of the fraction of circular polarization depending on the α and γ values, and on the presence of linear polarization intrinsic to the source. The second- and higher-orders discrepancies between measured and actual circular polarization introduced by α and γ are shown in Figs. 3 and 4, respectively. Note that discrepancies introduced by α deviations are those that have the largest impact on the measured P Q value Linear and circular polarization: effects due to h We finally discuss the effects of an imperfectly calibrated difference in the transmission function between parallel and perpendicular beams (e.g., an imperfect flat-fielding correction, or a dependence of the grism or CCD transmission functions on the polarization status of the radiation). P (D) Figure 5 shows the difference between the values obtained adopting the difference method and the ideal value P, as function of h. For small P values, the error introduced by h may be considered practically consistent with that due to photon noise, but in general will lead to a systematic underestimate of the actual polarization. In contrast, the ratio method is independent of h to any order. An important consequence of this result is that, in the context of the ratio method, any element that follows the Wollaston prism (unless it introduces severe problems of scattered light) will not affect the polarimetric measurements, although of course it may influence the signal-to-noise ratio obtained in given time. 6. The null parameters Null profiles were originally introduced, in the context of the ratio method, by Donati et al. (1997). In this work we have generalized the concept of null parameters to the context of the difference method, and we have analytically justified their importance as a diagnostic tool, showing that null parameters can be directly compared against their expected values, which are zero or close

21 21 to zero, with a Gaussian distribution characterized by the same FWHM as the Stokes parameters. In this Section we will consider the uniqueness of their definition, and we will further discuss the use of the null profiles for checking the quality of the reduced polarimetric data Alternative definitions of the null parameters The definition of null parameters is not unique. In the context of the difference method, the ideal quantities N (D ) = 1 N [ ] G [2k 1] + G [2k] ( 1) k 1 (56) 2N k=1 are still distributed about zero with the same standard deviation as the corresponding Stokes parameter. However, when considering the real null parameters, the first order expansions of the expressions of N (D ) given by Eq. (56) are different from 0 in all three cases = Q, U, and V, and higher (in absolute value) than those obtained with the definition of Eq. (38). In particular they include the term 1 2δh that may be potentially of the order of 1-2 %. Therefore, the definition of Eq. (56) would be a poorer choice with respect to our original definition of Eq. (20). ( ) Within the ratio method, the terms L [k] = r [2k 1] /r [2k] Ck used in Eqs. (26) may be replaced by the terms In the real case, to first order we obtain ( L [k] = r [2k 1] ) r [2k] Ck. (57) N (R ) Q = 1 2 P V δγ 1 PQ 2 N (R ) U = 1 P V 2 δγ 1 PU 2 N (R ) V = 0. (58) In the following we will continue to use our original definition of Eq. (26), although there is no obvious reason that should prevent one from using the definition based on Eq. (57) Quality checks with the null parameters Individual null parameter values should not be considered as error estimates, as their meaning has nothing to do with the statistical meaning of the classical error bar. Instead, it is most useful to compare the null parameter values with their expected σ, and for instance, rule out the results for which they are > 3 σ. In spectropolarimetric observations, it makes sense to compare the wavelength distribution of the null profiles with the corresponding Stokes spectra. The amplitude of the null spectrum will be representative of the noise of the Stokes profiles, while systematic deviations from zero may reflect the presence of a spurious polarization signal. For example, Eqs. (38) and (43)

22 22 show that cross-talk from circular polarization to Stokes Q due to a deviation of the nominal value of retardance phase of the half-wave plate has the same expression as the null profiles or N (R) Q ; thus, if one observes, in a linearly polarized source, a circular polarization profile P V similar to, the measured circular polarization can be probably ascribed to a cross-talk problem. N (D) Q While accurate absolute wavelength calibration may not be mandatory in spectropolarimetry, even a small wavelength shift between a spectral line observed in orthogonal polarizations may potentially lead to the detection of a spurious non-zero Stokes profile. This problem is documented in the leftmost panel of Fig. 6, which shows the differences between two Gaussian profiles shifted by 0.1 their FWHM. In fact, a systematic shift in wavelength calibration between parallel and perpendicular beam is compensated by the data reduction techniques discussed in this paper, and the same is true for a systematic shift between observations obtained at different positions of the retarder waveplate. However, a spurious selective shift between parallel and perpendicular beam, that affects data obtained at only one position of the retarder waveplate, will not be compensated, and will produce a signal of spurious polarization. These concepts are more clearly illustrated in the following example. To be as close as possible to the case of real observations, let us suppose that we have measured the circular polarization of a spectral line using four exposures taken at 45, +45, +45, and 45. This series of exposures is probably the most commonly adopted by observers (e.g., Donati et al. 1997; Wade et al. 2000; Bagnulo et al. 2006a). 1 Let us denote with f(λ 0 ) a spectral point well calibrated in wavelength, and with f(λ 0 + δλ) a spectral point that is shifted by a quantity δλ due to an imperfect wavelength calibration. Let us consider the two following situations: N (D) Q f 45 (λ 0 + δλ), f 45 (λ 0 + δλ), f +45 (λ 0), f +45 (λ 0) (a) f 45 (λ 0 + δλ), f 45 (λ 0), f +45 (λ 0 + δλ), f +45 (λ 0) (b). Situation (a) corresponds to a systematic shift introduced in both the parallel and perpendicular beams after a rotation of the retarder waveplate (e.g., due to the fact that the surface of the retarder waveplate is not perfectly perpendicular to the incident beam). Situation (b) describes a systematic shift in wavelength introduced between parallel and perpendicular beams by the Wollaston prism. While these shifts may be in principle calibrated if they are reproduced during the acquisition of an arc frame, both the difference method and ratio method will fully compensate for them, even if they have not been properly corrected during wavelength calibration (for a spectral line that is symmetric about the line center). 1 We note that, in the context of this Section, the values of the position angles denote different instrument settings that may lead to different artificial offsets of the spectral line profiles, whereas spurious polarization signals due to deviations of the polarimetric optics from the nominal values are not considered.

23 23 None of the methods will cancel spurious polarization if an instrumental wavelength shift appears in only one beam, as described by f 45 (λ 0 + δλ), f 45 (λ 0), f +45 (λ 0), f +45 (λ 0), (c). In the most general case, one may expect a different wavelength shift associated to each beam: f 45 (λ 0 + λ ), f 45 (λ 0 + λ ), f +45 (λ 0 + λ + ), f +45 (λ 0 + λ + ) (59) This situation always leads to a spurious polarization signal in spectral lines, and should be strenuously avoided. Ironically, the situation of Eq. (59) is most likely to occur when adopting the wavelength calibration strategy that, at a first sight, seems the most appropriate one to avoid it, namely, if one associates the scientific frames taken at different positions of the retarder waveplate with the arc frames obtained at the corresponding positions of the retarder waveplate. For instance, after a series of scientific exposures obtained with the retarder waveplate at 45 and +45, the calibration plan of the FORS1 instrument of the ESO VLT provides an arc frame obtained at 45 and one at +45. A similar calibration may be naturally requested by any user of any instrument, as one is tempted to wavelength calibrate the science frames obtained with the retarder waveplate at 45 with the arc frame obtained at 45, and the science frames obtained with the retarder waveplate at +45 with the arc frame obtained at +45 (e.g. Bagnulo et al. 2002). Following this procedure, there are four independent solutions to the wavelength calibration, which may well fall in the situation of Eq. (59) due to the noise intrinsic to the calibration procedure. In fact, it is reasonable to assume that: i) if the retarder waveplate does not rotate perfectly perpendicularly to the incident beam, a shift between the beams obtained at 45 and +45 may be introduced, but that this shift is the identical in the parallel and perpendicular beams; ii) if a shift between parallel and perpendicular beams is introduced by the Wollaston prism, this shift is independent of the position of the retarder waveplate. Then, adopting an arc frame obtained at only one position of the retarder waveplate for the wavelength calibration, irrespective of the position of the retarder waveplate in the science frames, allows us to impose on Eq. (59) the constraint λ λ = λ + λ + (d). Very simple numerical experiments show that, under this assumption, the signal of spurious polarization is strongly reduced compared to the situation in which all shifts are independent of each other. Examples of situations (a) to (d) are shown in the right panels of Fig. 6, where we have considered an arbitrary shift between parallel and perpendicular beam for situations (a) and (b), which are fully compensated by the data reduction techniques, and an artificial shift equal to times the FWHM between the profiles measured in the parallel and the perpendicular beams in only one exposure, to illustrate situation (c). For situation (d) we have set λ λ = λ + λ + = 0.01 FWHM, and λ + λ = 0.01 FWHM.